Recursive marginal quantization of higher-order schemes

Abstract

Quantization techniques have been applied in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and efficient calibration of large derivative books. Recursive Marginal Quantization of the Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This method involves recursively quantizing the conditional marginals of the discrete-time Euler approximation of the underlying process. By generalizing this approach, we show that it is possible to perform recursive marginal quantization for two higher-order schemes: the Milstein scheme and a simplified weak order 2.0 scheme. As part of this generalization a simple matrix formulation is presented, allowing efficient implementation. We further extend the applicability of recursive marginal quantization by showing how absorption and reflection at the zero boundary may be incorporated, when this is necessary. To illustrate the improved accuracy of the higher order schemes, various computations are performed using geometric Brownian motion and its generalization, the constant elasticity of variance model. For both processes, we show numerical evidence of improved weak order convergence and we compare the marginal distributions implied by the three schemes to the known analytical distributions. By pricing European, Bermudan and Barrier options, further evidence of improved accuracy of the higher order schemes is demonstrated.

Option Pricing with Orthogonal Polynomial Expansions

“…Our results improve the option pricing performance in Ackerer et al (2018). Notable progress has recently been made on the pricing of early-exercise and path-dependent options with stochastic volatility using recursive marginal quantization as studied in Pagès and Sagna (2015); McWalter et al (2017); Callegaro et al (2017a). In the context of affine and polynomial models, this approach has been shown to perform well when combined with Fourier transform techniques as in Callegaro et al (2017b), or polynomial expansion techniques as in Callegaro et al (2017), whose results could be further improved with the new expansions presented in our paper.…”

Quantization Under the Real-world Measure: Fast and Accurate Valuation of Long-dated Contracts

“…In this section, recursive marginal quantization is used to provide fast and accurate pricing for the benchmark approach. RMQ was introduced by Pagès and Sagna [2015], and extended to higher-order schemes by McWalter et al [2018].…”

Fast Quantization of Stochastic Volatility Models

“…In this section a concise matrix formulation for the JRMQ algorithm is presented, similar to that provided in McWalter et al [2017] for the standard RMQ case.…”

“…For each strike, computing an option price using Monte Carlo simulation takes approximately 14.5 seconds for the European options and 16.9 seconds for the Bermudan options. The high-level algorithm for pricing Bermudan options using a quantization grid is outlined in McWalter et al [2017].…”

“…Note that the distortion function is applied element-wise to the column vector (n) Γ x , and (0) Γ x is an initial guess for the quantizer. McWalter et al [2017] provide explicit expressions for the gradient vector and Hessian matrix in the one-dimensional case, and an efficient matrix formulation for implementation.…”